A Perturbation of the Dunkl Harmonic Oscillator on the Line

作者：Alvarez Lopez Jesus A; Calaza Manuel

来源：Symmetry Integrability and Geometry-Methods and Applications, 2015, 11: 059.

DOI：10.3842/SICMA.2015.059

• Dunkl harmonic oscillator
• perturbation theory

Let J(sigma) be the Dunkl harmonic oscillator on R (sigma > -1/2). For 0 < u < 1 and xi > 0, it is proved that, if sigma > u - 1/2, then the operator U = J(sigma) + xi vertical bar x vertical bar(-2u), with appropriate domain, is essentially self-adjoint in L-2 (R,vertical bar x vertical bar(2 sigma)dx), the Schwartz space S is a core of (U) over bar (1/2) and (U) over bar has a discrete spectrum, which is estimated in terms of the spectrum of (J) over bar (sigma). A generalization J(sigma),(tau) of J(sigma) is also considered by taking different parameters sigma and tau on even and odd functions. Then extensions of the above result are proved for J(sigma,tau), where the perturbation has an additional term involving, either the factor x(-1) on odd functions, or the factor x on even functions. Versions of these results on R+ are derived.

• 出版日期2015